Study Problems on Market Risk
MF 830
Not for handing in
Solutions
1. Should financial institutions with illiquid positions use VAR to measure their risk?
The premise of VAR is that you are stuck with a given set of positions for a while, and VAR tells
you the ‘maximum’ loss under a given confidence level. So, if a market is illiquid, it means that
you might be stuck holding the positions for a long time because it is hard to trade out of them.
Thus, VAR is OK in principle for illiquid positions as long as you build the statistics assuming a
long holding period.
2. What are the two main weaknesses of the delta-normal model?
Delta normal assumes that the underlying risk factor has a normal distribution and that the
relationship between your portfolio value and the risk factor is linear. Many financial prices
have distributions whereby large outliers happen more frequently than they would under a
normal distribution, which implies that the delta-normal method may provide too-small estimates
of VAR since large changes happen more frequently than the model expects. Also, many
positions are non-linear functions of the risk factor. For example, a bond is a non-linear function
of the interest rate since it has positive convexity. The linearity assumption works well for small
changes in the risk factor and badly for large changes, as we saw when we studied interest rate
risk. This means that for longer holding periods, where larger changes in the risk factor become
more common, delta-normal will tend to have a greater error.
3. Some banks compute VAR using a statistical procedure placing greater weight on data
from the recent past. Do you think this is a good idea or a bad idea? Related questions:
What is the advantage of a long estimation period? What is the downside of a long
estimation period?
Putting more weight on the recent past maximizes the chance that the VAR model will work well
over the next day. The reason this is true is that financial markets tend to be persistently volatile
or non-volatile. Hence, if you use recent data your estimates of things like the standard deviation
of returns will tend to accurately reflect current market conditions. The problem with this
approach is that such models can fail spectacularly when the ‘Black Swan’ events happen. That
is, when markets shift suddenly from very calm to very volatile, as might happen after a major
bank failure or war, your model will dramatically understate risk. This answer really gets at the
essence of the tradeoff between a long v. short estimation period. Long estimation periods will
tend to pick up data across all kinds of market conditions; hence it will never be too bad.
However, the long estimation strategy will not be as sensitive to recent market conditions, which
Page 1 of 4
as I said earlier tend to persist. Hence, the long estimation period VAR will be less accurate
most of the time (but more accurate – or at least less bad – when market conditions change).
4. What is the single most fundamental assumption that justifies the use of VAR? What can
you do if this assumption is not correct?
All models make the fundamental assumption that tomorrow (i.e. the holding period) will look
like yesterday (the estimation period). Again, this is the ‘Black Swan’ problem. Models will
never be able to anticipate events that are truly new or revolutionary. Does this mean you should
throw out the model? No, because Black Swans (wars, big bank failures, etc) are rare events. So
models work very well most of the time. The way to deal with these big events is: 1) recognize
that they may happen; 2) realize that they are not accounted for by models; 3) run the portfolio
through stress tests or scenarios that could lead to big losses to understand at least the potential
consequences of such bad events on your firm or your portfolio.
5. JP Morgan’s VAR changed substantially during 2008. What are some possible
explanations for why a bank’s VAR might vary over the year?
First, Morgan bought Bear in 2008 and took on their positions. Large changes in position size
will tend to change VAR. To the extent that adding Bear scaled up many of Morgan’s positions,
this would tend to lead to a higher VAR. Second, market conditions changed dramatically
during the year. Morgan’s VAR technology places greater weight on recent data (see Q3 above),
and is therefore quite sensitive to shifts in volatility. Hence, their VAR numbers would have
been much higher in the fourth quarter of 2008 than during the early quarters.
6. Imagine that you are managing a trading portfolio worth $50 million. The return on your
portfolio is normally distributed with an annual standard deviation of 25%. Compute the
1-day and 10-day VAR with 99% confidence. Discuss when you would use the 1-day
VAR to manage market risk, and when you would prefer the 10-day VAR.
We just need to do the root t ‘rule’ in reverse, which means divide rather than multiply.
One-day VAR = 2.33 ($50) (0.25) / √250 = $1.84 (Since 25% is annualized)
Ten-day VAR = 1.84 * √10 = 5.825
If markets are very liquid, go with one day VAR. If you are worried about lack of liquidity, but think you
could trade out of the position efficiently over a two-week span, go with the 10-day VAR.
Page 2 of 4
7. You are running a hedge fund with a long position of 1000 shares of IBM, and a short
position of 5000 shares of Intel. IBM is currently trading at $120 per share, and Intel is
trading at $22 per share. Over the past year, the volatility of IBM=s stock was 2% per
day, the volatility of Intel=s stock was 3% per day, and the correlation between the two
company stocks was 0.9. Compute the delta-normal VAR with a 1-day holding period
and 99% confidence for your portfolio. Suppose your hedge fund=s capital (i.e. its net
worth) equals the value of its current positions (e.g. it has no other assets). Based on your
calculation, does your hedge fund hold enough capital? During the past year, there was
one day in which IBM=s shares fell 4%. On that same day, Intel=s shares rose by 6%.
How does this change your answer about capital adequacy? What does this illustrate
about hedges that are based on correlations?
VaR of IBM = 2.33*1000*120*0.02 = 5592
VaR of Intel = 2.33*5000*22*0.03 = 7689
Portfolio VaR = (55922 + 76892 - 2*0.9*5592*7689)1/2 = 3605
The fund=s capital = 1000*120 - 5000*22 = 10,000. Given the potfolio VAR, this is
plenty of capital. (That is, assuming the assumptions are correct...)
The day in which the two stocks diverged in price shows that the portfolio risk is actually
very high. Just one day where the prices move in opposite direction buries the hedge!!
We lose all of our capital: the loss = $11,400. The point here is that the hedge relies on
the very high positive correlation in the risk factors. Correlations are notoriously
unstable!
Page 3 of 4
8. Imagine that you are managing a trading portfolio with 100 shares of stock in Samsung.
The current price is 12,800 Won (the South Korean currency). The standard deviation of
the daily return on Samsung estimated over the past 3 months is 1 percent. Use delta
normal to compute the 1-day, 99% VAR for a South Korean bank (i.e. in Won). The
$/Won exchange rate is currently 1/980 (that is $1 = 980 Won), and the standard
deviation of the exchange rate (in percentage terms) is 0.4% per day, again using the past
3 months of data. Also, the correlation between the percentage change in the exchange
rate and the return on Samsung is estimated to be equal to 0.5. Use delta normal to
compute the 1-day, 99% VAR for the same trading portfolio from the perspective of a US
bank (i.e. in dollars). During past three months, the $/Won exchange rate decreased by 2
percent (this was the largest absolute change during the period). Also, the standard
deviation of the percentage change in the exchange rate was 1% per day during the first
three months of last year. How do these additional facts affect your assessment of risk if
you are running a US bank? What other risk measures would you like to see?
The Korean bank cares about VAR in Won:
VAR from price movement = 2.33*100*12,800*0.01 = 29,824 won
In this example, 100 is the delta; the level of price is 12800; the volatility of the risk factor is 1%.
The US bank faces two risk factors:
VAR from price movement, converted to dollars = 29,824 won = $30.43
VAR for F/X movement = 2.33*(100*12,800)(1/980)*0.004 = $12.17
In this case, (100*12,800) is the delta; the level of the risk factor is 1/980; 0.4% is the volatility.
So, total VAR for the US bank = √(30.432+12.172+2(0.5)*30.43*12.17) = $38.01
The additional facts make me worry about using 0.4% as the volatility for exchange rates in a
delta normal model. First, it would be very unlikely to observe a 5 standard deviation move in
the exchange rate if it were really normally distributed with a standard deviation of 0.4%.
Second, the fact that using a different estimation period gives such a big difference in the
estimate of the F/X volatility is also a cause for concern. To deal with the non-normality
problem, I might want to construct VAR under historical simulation. To deal with uncertainty
about true F/X volatility, I might want to build VAR using several different estimation periods.
Page 4 of 4